How can we prove the universe is expanding?
Using the red shift of the doppler effect.
The observations that most of the universe exhibits a red shift in the spectrum. Objects moving away from an observed exhibit a red shift. This is because as the object moves away the waves are spread further apart. As the waves move further apart the wavelength becomes longer which moves the spectrum toward the red side of the spectrum.
The red shift as explained by the doppler effect shows that the universe is expanding.
The observations of Hubble about the red shift caused a shift in the philosophical theories of the universe. Before Hubble's observations the accepted theory was the steady state. The idea was that the universe was eternal and had always existed in its present state. Hubble's observations of the red shift which implies an expanding universe destroyed the steady state universe theory.